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Question 1 of 6
Find the derivative using the chain rule
5cos4x
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
First, identify the values of the function
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
5⋅1⋅cos(4x)⋅f′(4x) |
Substitute known values |
|
|
= |
5⋅(−sin4x)⋅4 |
Differentiate the values |
|
|
= |
−20sin4x |
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Question 2 of 6
Find the derivative using the chain rule
tan(4x-2)
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
First, identify the values of the function
f(x) |
= |
xn |
f(x) |
= |
tan(4x−2) |
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
1⋅tan(4x−2)⋅f′(4x−2) |
Substitute known values |
|
|
= |
(sec2(4x−2)⋅4 |
Differentiate the values |
|
|
= |
4sec2(4x−2) |
Simplify |
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Question 3 of 6
Find the derivative using the chain rule
√cosx
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
First, convert the surd into an exponent
Next, identify the values of the function
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
12⋅(cosx)(12)−1⋅f′(cosx) |
Substitute known values |
|
|
= |
12(cosx)−12⋅−sinx |
Differentiate cos x |
|
|
= |
−sinx2(cosx)12 |
Reciprocate cos x-12 |
|
|
= |
−sinx2√cosx |
Convert the exponent into a surd |
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Question 4 of 6
Find the derivative using the chain rule
sin4x
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
The expression can also be written as
Next, identify the values of the function
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
4⋅(sinx)4−1⋅f′(sinx) |
Substitute known values |
|
|
= |
4(sinx)3⋅cosx |
Differentiate the values |
|
|
= |
4sin3xcosx |
Evaluate |
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Question 5 of 6
Find the derivative using the chain rule
cos3 2x
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
The expression can also be written as
Next, identify the values of the function
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
3⋅(cos2x)3−1⋅f′(cos2x) |
Substitute known values |
|
|
= |
3(cos2x)2⋅−sin2x⋅2 |
Differentiate the values |
|
|
= |
−6cos22xsin2x |
Evaluate |
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Question 6 of 6
Find the derivative using the chain rule
3 sin(5x-3)
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Derivatives of Trigonometric Functions
Chain Rule
y′=n⋅(f(x))n−1⋅f′(x)
First, identify the values of the function
f(x) |
= |
xn |
f(x) |
= |
3sin(5x−3) |
Finally, substitute the values into the chain rule
y’ |
= |
n⋅(f(x))n−1⋅f′(x) |
|
|
= |
3⋅1⋅sin(5x−3)⋅f′(5x−3) |
Substitute known values |
|
|
= |
3(cos(5x−3))⋅5 |
Differentiate the values |
|
|
= |
15cos(5x−3) |
Evaluate |